An Odd Couple: Art and Math

January 2019’s topic of The GalleseumAcrylic Math’s FREE monthly art newsletter– is about the relationship between art and math, which dates back to antiquity and spans to modern times. And, this is the corresponding Blog Post.

An Odd Couple?
Art and Math

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Blog Post

An odd couple? Not really! The relationship between art and math dates back to antiquity and spans to modern times. In 4 BC, the Greek sculptor Polykleitos of Argos described the ideal mathematical proportions of the human body in a work titled the Kanon. [1] [2] During the Renaissance, Leonardo da Vinci, the Italian genius, also described the ideal mathematical proportions of the human body in a drawing titled L’Uomo Vitruviano [3] (See Figure 1: The Vitruvian Man by Leonardo da Vinci. Public domain. [4])

The Vitruvian Man

And, in modern times, Piet Mondrian, the famous Dutch painter, used simple geometric elements in his work [5] (See Figure 2: Composition No. III by Piet Mondrian. Public domain. [6]).

Figure 2

The Golden Ratio

The golden ratio is represented by the Geek letter phi (φ), is equal to 1.618034…, and was defined by Euclid, the father of geometry, in a work titled Elements. [7] Oftentimes, the golden ratio is displayed as the golden rectangle, whose sides are equal to 1: φ [8] (See Figure 3: Golden rectangle. Public domain. [9]).

Figure 3

Art cognoscenti have identified the use of the golden rectangle in design. For example, Samuel Obara of the Department of Mathematics of the University of Georgia recognized some φ-based rectangles in Piet Mondrian’s Composition No. II [10] (See Figure 4: Composition No. II by Piet Mondrian. Public domain. [11]).

Figure 4

Tessellations

Tessellations, from the Latin tessella (small square), are tilings of continuous shapes: Euclidean, organic, and three-dimensional. [12] Figure 5 (Penrose tiling. Public domain. [13]) displays an example of a tessellation.

Figure 5

Tesselations were used in ancient Rome and in the Islamic world, notably in the Alhambra, in Granada, Spain (See Figure 6: Tessellation, Alhambra, Seville, Spain. © 2007 Gruban. Reprinted with permission. [14]) In modern times, the renowned Dutch artist M.C. Escher use tessellations in his work. [12]

Figure 6

Fractals

Fractals, from the Latin fractus (broken), are detailed patterns that endlessly repeat themselves at different scales. Fractals are characterized by self-similarity and non-integer dimensions. [15] [16] Fractal geometry is rooted in the seminal works of Gottfried Leibniz, the 17th century German philosopher and mathematician, and others that followed, notably Benoit Mandelbrot, the 20th century Polish-born mathematician. [17] Figure 7 (Triangle fractal. © Fractal Foundation. Reprinted with permission. [16]) displays an example of a fractal.

Figure 7

Summary

Myriad examples document the relationship between art and math. The three topics briefly presented herein -the golden ratio, tessellations, and fractals- are but “small cogs in the large wheel” of art and math.

Endnotes

[1] Mathematics and art. (2018). From Wikipedia. URL: https://en.wikipedia.org/wiki/Mathematics_and_art

[2] Polyclitus. (2018). From Encyclopedia Britannica. URL: https://www.britannica.com/biography/Polyclitus

[3] Vitruvian Man. (2018). From Wikipedia. URL: https://en.wikipedia.org/wiki/Vitruvian_Man

[4] Da Vince, L. (2018). Vitruvian Man. From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:Da_Vinci_Vitruve_Luc_Viatour.jpg

[5] Jaffe, H.L.C. (2018). Piet Mondrian. From Encyclopedia Britannica. URL: https://www.britannica.com/biography/Piet-Mondrian

[6] Mondrian, P. (1929). Composition No. III. From The Athenaeum. URL: https://www.the-athenaeum.org/art/detail.php?ID=85852

[7] Berry, B. (2017). What is the golden ratio? From Math Hacks. URL: https://medium.com/i-math/what-is-the-golden-ratio-d3cc17c8fefd

[8] Golden rectangle. (2018). From Wikipedia’s. URL: https://en.wikipedia.org/wiki/Golden_rectangle

[9] Golden Rectangle. (2018). From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:SimilarGoldenRectangles.svg

[10] Obara, S. (n.d.). Golden ratio in art and architecture. From The University of Georgia, URL: http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html

[11] Mondrian, P. (1030). Composition No. II. From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:Piet_Mondriaan,_1930_-_Mondrian_Composition_II_in_Red,_Blue,_and_Yellow.jpg

[12] Tessellation. (2018). From Wikipedia. URL: https://en.wikipedia.org/wiki/Tessellation

[13] Penrose tiling. (2009). From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:Penrose_Tiling_(P1_over_P3).svg

[14] Gruban. P. (2007). Tesselletion, Alhambra, Seville, Spain. From Wikimedia Commons. URL: https://commons.wikimedia.org/wiki/File:Tassellatura_alhambra.jpg

[15] Fractal. (2018). From Encyclopedia Britannica. URL: https://www.britannica.com/science/fractal

[16] What is a fractal? (n.d.) From Fractal Foundation. URL: https://fractalfoundation.org/fractivities/WhatIsaFractal-1pager.pdf

[17] Fractal. (2018). From Wikipedia.URL: https://en.wikipedia.org/wiki/Fractal